If you add cream of tartar, be sure to dilute it with an equal amount of water before adding it. It is an acid, and has the effect of causing like poles of sugar to spin around towards each other, much like trying to force like poles of 2 magnets together, and thereby inhibiting re-crystalization. Alternatively, add a bit of corn syrup or honey, which are already invert sugars, and which already contain a small amount of acid to keep them that way. The amount is proportional to the amount of caramel you're making, but a couple of ounces should do the trick for you.

Couple of things to try: make sure the water is covering the sugar when you start out. I don't stir or swirl; I just let it come to a boil and brush down the sides. The swirling should begin when you start to see a light golden color begin to form. the swirling helps the sugar turn color evenly. Do it gently. I also find that a copper pot or a heavy bottom, medium pot works best.